J-Stability in non-archimedean dynamics

TL;DR

This paper investigates the stability of Julia sets in non-archimedean dynamics, showing that under certain contraction conditions, small perturbations preserve the conjugacy of dynamics.

Contribution

It establishes a stability result for Julia sets of rational functions over non-archimedean fields under bounded contraction conditions.

Findings

Small perturbations preserve Julia set dynamics

Conjugacy of dynamics is maintained under contraction conditions

Results apply to rational functions over complete, algebraically closed non-archimedean fields

Abstract

Let $C_v$ be a complete, algebraically closed non-archimedean field, and let $f \in C_v(z)$ be a rational function of degree $d \geq 2$. If $f$ satisfies a bounded contraction condition on its Julia set, we prove that small perturbations of $f$ have dynamics conjugate to those of $f$ on their Julia sets.